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Tropical Geometry and Mirror Symmetry

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Mark Gross

A co-publication of the AMS and CBMS

Tropical geometry provides an explanation for the remarkable power of
mirror symmetry to connect complex and symplectic geometry. The main
theme of this book is the interplay between tropical geometry and mirror
symmetry, culminating in a description of the recent work of Gross and
Siebert using log geometry to understand how the tropical world relates
the A- and B-models in mirror symmetry.

The text starts with a detailed introduction to the notions of
tropical curves and manifolds, and then gives a thorough description
of both sides of mirror symmetry for projective space, bringing
together material which so far can only be found scattered throughout
the literature. Next follows an introduction to the log geometry of
Fontaine-Illusie and Kato, as needed for Nishinou and Siebert's proof
of Mikhalkin's tropical curve counting formulas. This latter proof is
given in the fourth chapter. The fifth chapter considers the mirror,
B-model side, giving recent results of the author showing how tropical
geometry can be used to evaluate the oscillatory integrals appearing.
The final chapter surveys reconstruction results of the author and
Siebert for “integral tropical manifolds.” A complete
version of the argument is given in two dimensions.

A co-publication of the AMS and CBMS.

Readership

Graduate students and research mathematicians interested in
mirror symmetry and tropical geometry.

Reviews & Endorsements

This book is well-written and
provides very useful introductory accounts of many aspects of the highly
involved subject of mirror symmetry. This book can be extremely helpful to
those who want to understand mirror symmetry and the Gross-Siebert program.

Affiliation(s) (HTML):
University of California, San Diego, San Diego, CA

Publisher Blurb:
A co-publication of the AMS and CBMS

Abstract:

Tropical geometry provides an explanation for the remarkable power of
mirror symmetry to connect complex and symplectic geometry. The main
theme of this book is the interplay between tropical geometry and mirror
symmetry, culminating in a description of the recent work of Gross and
Siebert using log geometry to understand how the tropical world relates
the A- and B-models in mirror symmetry.

The text starts with a detailed introduction to the notions of
tropical curves and manifolds, and then gives a thorough description
of both sides of mirror symmetry for projective space, bringing
together material which so far can only be found scattered throughout
the literature. Next follows an introduction to the log geometry of
Fontaine-Illusie and Kato, as needed for Nishinou and Siebert's proof
of Mikhalkin's tropical curve counting formulas. This latter proof is
given in the fourth chapter. The fifth chapter considers the mirror,
B-model side, giving recent results of the author showing how tropical
geometry can be used to evaluate the oscillatory integrals appearing.
The final chapter surveys reconstruction results of the author and
Siebert for “integral tropical manifolds.” A complete
version of the argument is given in two dimensions.

A co-publication of the AMS and CBMS.

Book Series Name:
CBMS Regional Conference Series in Mathematics

Volume:
114

Publication Month and Year:
2011-01-20

Copyright Year:
2011

Page Count:
317

Cover Type:
Softcover

Print ISBN-13:
978-0-8218-5232-3

Online ISBN 13:
978-1-4704-1572-3

Print ISSN:
0160-7642

Online ISSN:
0160-7642

Primary MSC:
14;
52

Textbook?:
False

Applied Math?:
False

Electronic Media?:
False

Apparel or Gift:
False

Publisher (non-AMS):
A co-publication of the AMS and Conference Board of Mathematical Sciences

Graduate students and research mathematicians interested in
mirror symmetry and tropical geometry.

Reviews:

This book is well-written and
provides very useful introductory accounts of many aspects of the highly
involved subject of mirror symmetry. This book can be extremely helpful to
those who want to understand mirror symmetry and the Gross-Siebert program.